Drill string axial vibration attenuator

ABSTRACT

The drill string axial vibration attenuator ( 10 ) is installed in the bottom hole assembly (BHA) of a drill string to attenuate axial and torsional vibrations of the drill string. The vibration attenuator ( 10 ) includes a massive elongate stabilizer ( 14 ) installed in a sealed chamber ( 12 ) that is, in turn, rigidly installed within the BHA of the drill string. Clearance between the inner wall of the chamber ( 12 ) and the stabilizer ( 14 ) is provided to preclude frictional interference therebetween. The stabilizer mass ( 14 ) is supported from below by a compression spring ( 18 ) and a shock absorber or damper ( 24 ) at the top to allow movement of the mass ( 14 ). The stabilizer ( 14 ) substantially reduces or eliminates axial and coupled torsional vibration of the drill string when the mass of the stabilizer ( 14 ), the rate of the spring, and the damper stiffness are properly configured.

TECHNICAL FIELD

The present invention relates generally to earth boring and drillingequipment, and particularly to a drill string axial vibration attenuatorfor damping or attenuating undesired axial motion in a drill stringduring drilling operations.

BACKGROUND ART

The problem of drill string vibration has been recognized as one of theprime causes of deterioration in drilling performance. These vibrationsmay be lateral, torsional, or axial. Field observations have indicatedthat drill strings can exhibit severe vibrations that may become evenmore severe at the bottom-hole assembly (BHA), which comprises the drillcollars, stabilizers, and the drill bit, and may also include otherlogging tools and instruments.

This application is directed to axial and torsional drill stemvibrations, and means for reducing or eliminating such axial andtorsional vibrations. As the drill bit is penetrating the underlyingformation during the drilling operation, the normal reaction force orweight-on-bit (WOB) may become excessive and fluctuate, resulting inaxial vibration in the drill string. This is known as “bit-bounce.”Excessive axial vibration or bit-bounce may result in reduction of therate of penetration (ROP) of the drill bit and/or damage to the drillbit, adverse effects upon telemetry tools and data conveyed to thesurface, and fatigue of the drill pipes that form the drill stem. All ofthese factors result in decreased efficiency in the drilling process andincreased costs of operation due to the need to replace variouscomponents more frequently than would be the case without such axialdrill string vibrations.

Excessive torsional vibrations may eventually result in limit cycleswhere the BHA rotary speeds are bounded between zero and two or possiblyeven three times the designated rotary table (rotary drive for the drillstem) speed. At its extreme, this phenomenon is known as “stick-slip”where the rotational velocity of the drill bit is momentarily decreasedto zero as it sticks at the bottom of the down hole, then slips andaccelerates beyond the prescribed rotary table speed. This “stick-slip”phenomenon is also detrimental to the drill pipes and string, the drillbit, logging tools, and to the entire drilling operation.

Thus, a drill string axial vibration attenuator solving theaforementioned problems is desired.

DISCLOSURE OF INVENTION

The drill string axial vibration attenuator comprises a massive,elongate stabilizer installed concentrically within a sealed internalchamber of the drill collar (DC) or BHA of a drill string, the chamberbeing rigidly secured concentrically within the DC or BHA. Drillingfluid or “mud” is routed around the sealed stabilizer chamber betweenthe outer wall of the chamber and the inner wall of the drill stringpipe. The stabilizer is supported within its chamber by a compressionspring at its lower end and by a shock absorber or damper extendingbetween the upper end of the stabilizer and the drill string structure.Clearance is provided around the stabilizer to preclude contact with thesurrounding wall of the sealed chamber.

The mass of the suspended stabilizer, the spring rate of the supportingcompression spring, and/or the stiffness of the damper may be adjustedor configured according to the needs of the system. The spring anddamper rates may be linear or nonlinear. The stiffness, mass, and/ordamping values for the assembly are selected after considering the drillbit type, type of geological formation, DC and/or BHA mass, and/or otherdrill string parameters. Axial vibrations of the drill string aredissipated due to the damping forces that arise from the relativevelocity between the stabilizer mass and the DC and/or the BHA of thedrill string. This provides two beneficial effects, namely, (1) resonantand off-resonance forced vibrations are attenuated; and (2) vibrationinstabilities caused by modulation of cutting force amplitudes aresuppressed.

Torsional vibrations may also be attenuated by rotating or spinning thestabilizer within its chamber. A mathematical analysis of the variousrelevant parameters is also provided herein.

These and other features of the present invention will become readilyapparent upon further review of the following specification anddrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic elevation view of a drill string axial vibrationattenuator according to the present invention.

FIG. 2 is a schematic elevation view showing the drill string axialvibration attenuator of FIG. 1 installed in the bottom hole assembly(BHA) of the drill string.

FIG. 3A is a schematic diagram of a drill string and drive table,illustrating various parameters related to axial and torsional forcesexerted upon the drill string.

FIG. 3B is a schematic diagram of the drill string and drive table ofFIG. 3A, including a diagrammatic equivalent structure representation ofa drill string axial vibration attenuator according to the presentinvention.

FIG. 4A is another schematic diagram of the drill string of FIGS. 3A and3B, showing additional parameters affected by operation of a drillstring axial vibration attenuator according to the present invention.

FIG. 4B is a schematic diagram of a polycrystalline-diamond-compact(PDC) drill bit, illustrating various additional parameters of concernwhen the drill string axial vibration attenuator according to thepresent invention is used therewith.

FIG. 5A is a graph showing the minimum added mass required to suppressbit-bounce using the drill string axial vibration attenuator accordingto the present invention, wherein the rock stress is 160 MPa and theformation stiffness is 134 MPa.

FIG. 5B is a three-dimensional graph showing stable mass values for thedrill string axial vibration attenuator according to the presentinvention.

FIG. 6A is a graph of axial BHA velocity without the drill string axialvibration attenuator according to the present invention when rock stressis 160 MPa and formation stiffness is 134 MPa.

FIG. 6B is a graph of rotational BHA velocity without the drill stringaxial vibration attenuator according to the present invention when rockstress is 160 MPa and formation stiffness is 134 MPa.

FIG. 7A is a graph of axial BHA velocity with the drill string axialvibration attenuator according to the present invention installed in theBHA when rock stress is 160 MPa and formation stiffness is 134 MPa.

FIG. 7B is a graph of rotational BHA velocity with the drill stringaxial vibration attenuator according to the present invention installedin the BHA when rock stress is 160 MPa and formation stiffness is 134MPa.

FIG. 8 is a three-dimensional plot of stable values for the drill stringaxial vibration attenuator according to the present invention, asinstalled and operating within the BHA.

FIG. 9 is a graph of axial velocity oscillations of a conventional drillstring without the installation of the drill string axial vibrationattenuator according to the present invention therein.

FIG. 10A is a plot of Eigenvalues in a conventional drill string withoutthe installation of the drill string axial vibration attenuatoraccording to the present invention.

FIG. 10B is a plot of Eigenvalues for a drill string having the drillstring axial vibration attenuator according to the present inventioninstalled therein.

FIG. 11 is a coarse mesh stability chart for the drill string axialvibration attenuator according to the present invention.

FIG. 12A is a graph showing the axial velocity of the BHA of the drillstring incorporating the drill string axial vibration attenuatoraccording to the present invention at a top drive speed of 78 RPM andformation stiffness of 53 MPa.

FIG. 12B is a graph showing the axial velocity of the BHA of the drillstring incorporating the drill string axial vibration attenuatoraccording to the present invention at a top drive speed of 78 RPM andformation stiffness of 55 MPa.

FIG. 13A is a graph showing the drill bit rotational velocity of a drillstring incorporating the drill string axial vibration attenuatoraccording to the present invention at a top drive speed of 105 RPM andformation stiffness of 100 MPa.

FIG. 13B is a graph showing the axial velocity of the BHA of the drillstring incorporating the drill string axial vibration attenuatoraccording to the present invention at a top drive speed of 135 RPM andformation stiffness of 154 MPa.

FIG. 13C is a graph showing the drill bit rotational velocity of a drillstring incorporating the drill string axial vibration attenuatoraccording to the present invention at a top drive speed of 140 RPM andformation stiffness of 158 MPa.

FIG. 14 is an operating point stability chart showing variouscharacteristics for a drill string with and without the drill stringaxial vibration attenuator according to the present invention.

FIG. 15A is a graph showing the axial velocity of the lower or distalportion of a drill stem adjacent to the drill bit prior to theinstallation of the drill string axial vibration attenuator according tothe present invention.

FIG. 15B is a graph showing the torsional velocity of the lower ordistal portion of a drill stem adjacent to the drill bit prior to theinstallation of the drill string axial vibration attenuator according tothe present invention.

FIG. 15C is a graph showing the drill bit reaction force or weight onbit (WOB) prior to the installation of the drill string axial vibrationattenuator according to the present invention.

FIG. 15D is a graph showing the drill bit reaction torque, or torque onbit (TOB), prior to the installation of the drill string axial vibrationattenuator according to the present invention.

FIG. 16A is a graph showing the axial velocity of the lower or distalportion of a drill stem adjacent to the drill bit after installation ofthe drill string axial vibration attenuator according to the presentinvention.

FIG. 16B is a graph showing the torsional velocity of the lower ordistal portion of a drill stem adjacent to the drill bit afterinstallation of the drill string axial vibration attenuator according tothe present invention.

FIG. 16C is a graph showing the drill bit reaction force or weight onbit (WOB) after installation of the drill string axial vibrationattenuator according to the present invention.

FIG. 16D is a graph showing the drill bit reaction torque, or torque onbit (TOB), after installation of the drill string axial vibrationattenuator according to the present invention.

FIG. 17A is a graph showing the universal performance of the drillstring axial vibration attenuator according to the present inventionwith a polycrystalline-diamond-compact (PDC) drill bit over a range ofrotational speeds and a formation stiffness of 80 MPa.

FIG. 17B is a graph showing the universal performance of the drillstring axial vibration attenuator according to the present inventionwith a roller cone drill bit over a range of rotational speeds and aformation stiffness of 80 MPa.

FIG. 18A is a graph comparing the effects of sprung and unsprung massesto the BHA having the drill string axial vibration attenuator accordingto the present invention installed therein, using a PDC drill bit with aformation stiffness of 80 MPa.

FIG. 18B is a graph comparing the effects of sprung and unsprung massesto the BHA having the drill string axial vibration attenuator accordingto the present invention installed therein, using a roller cone drillbit with a formation stiffness of 80 MPa.

FIG. 18C is a graph comparing the effects of sprung and unsprung massesto the BHA having the drill string axial vibration attenuator accordingto the present invention installed therein, using a PDC drill bit withvariable formation stiffness and a top drive rotational speed of 80 RPM.

FIG. 18D is a graph comparing the effects of sprung and unsprung massesto the BHA having the drill string axial vibration attenuator accordingto the present invention installed therein, using a roller cone drillbit with variable formation stiffness and a top drive rotational speedof 80 RPM.

FIG. 19A is a graph comparing the axial velocities of a drill stringincorporating the drill string axial vibration attenuator according tothe present invention with various shock absorber or dampercharacteristics in a relatively soft rock formation.

FIG. 19B is a graph comparing the torsional velocities of a drill stringincorporating the drill string axial vibration attenuator according tothe present invention with various shock absorber or dampercharacteristics in a relatively soft rock formation.

FIG. 20A is a graph comparing the axial velocities of a drill stringincorporating the drill string axial vibration attenuator according tothe present invention with and without a shock absorber or damper in aharder rock formation.

FIG. 20B is a graph comparing the torsional velocities of a drill stringincorporating the drill string axial vibration attenuator according tothe present invention with and without a shock absorber or damper in aharder rock formation.

FIG. 21 is a graph showing the relationship between the mass of thestabilizer and the stable top drive rotational speed of a drill stringincorporating the drill string axial vibration attenuator according tothe present invention.

FIG. 22 is a graph showing the axial velocity at the BHA at 76 RPM in aconventional drill string without the installation of the drill stringaxial vibration attenuator according to the present invention.

FIG. 23 is a graph showing the axial velocity at the BHA at 76 RPM in adrill string incorporating the drill string axial vibration attenuatoraccording to the present invention when the attenuator has a mass of 20percent of the BHA, a natural frequency of 30 rad/sec, and a dampingratio of 0.3.

FIG. 24 is a graph showing the axial velocity with various top driverotational speeds of a BHA having a mass of 87 tons in a drill stringusing a PDC drill bit and incorporating the drill string axial vibrationattenuator according to the present invention when the actuator has anatural frequency of 30 rad/sec and a damping ratio of 0.3 and theformation stiffness is 134 MPa.

Similar reference characters denote corresponding features consistentlythroughout the attached drawings.

BEST MODES FOR CARRYING OUT THE INVENTION

The drill string axial vibration attenuator, also referred to herein asthe attenuator, is installed within a length of drill pipe in a drillstring to greatly reduce or eliminate vibrations in and along the drillstring during drilling operations. The attenuator is particularlyconfigured to dampen or eliminate axial vibrations, but may be used forthe reduction or elimination of torsional vibrations as well.

FIG. 1 of the drawings provides a schematic elevation view of the drillstring axial vibration attenuator 10. The attenuator 10 includes ahollow elongate chamber 12 that is sealed from the external environment.The chamber 12 is preferably in the form of an elongate cylinder, butother non-cylindrical shapes may be used, if desired. An elongatestabilizer mass 14 is installed concentrically within the chamber 12,the chamber 12 being immovably affixed within the drill pipe DP, asshown in FIG. 2. The stabilizer mass 14 is also preferably in the formof an elongate cylinder, but other non-cylindrical shapes may be used,if desired.

The stabilizer mass 14 is resiliently supported at its lower end 16 by acompression spring 18 that extends upward from within the lower end 20of the chamber 12. The opposite, upper end 22 of the stabilizer mass 14is connected to a shock absorber or damper 24 that extends downward fromthe upper end 26 of the chamber 12. The spring 18 may have a linear orconstant spring rate, or alternatively, may have a nonlinear rate, e.g.,having stiffer coils forming a portion of its length and lighter coilsforming the remainder of the length. Similarly, the damper 24 may have alinear or nonlinear damping rate.

The chamber 12 has an inner wall or surface 28 defining an internal spanor diameter that is larger than that of the stabilizer mass 14. Thus,the outer wall or surface 30 of the stabilizer mass 14 and the innerwall or surface 28 of the chamber 12 define a toroidal stabilizer massclearance volume 32 therebetween. Thus, the stabilizer mass 14 is freeof contact with the inner surface 28 of the chamber 12. Theabove-described structure enables the stabilizer mass 14 to movevertically, relative to the chamber 12, within the limits imposed by thespring 18 and damper 24. By selecting appropriate masses, spring rates,and damper rates according to the mass of the drill stem, or moreappropriately, to the mass of the drill pipe or bottom hole assembly inwhich the attenuator 10 is installed, the attenuator 10 serves to reduceor eliminate vertical vibrations along the drill stem between the drillbit and the top drive at the surface.

However, the drill stem is also subject to torsional vibrations due tofrictional drag at the drill bit and the relatively constant torque ofthe top drive. In extreme cases, the rotational velocity of the drillbit may be reduced to zero as the bit sticks in the substrate in whichit is working, and then accelerate to two or three times the rotationalspeed of the top drive when the bit is released due to the energy storedin the drill string when the bit sticks and the inherent torsionalelasticity of the elongate drill string. This phenomenon is known as“stick-slip” in the drilling industry.

The additional mass of the stabilizer 14 disposed within the drill pipecan serve to reduce or eliminate such torsional vibrations, even if thestabilizer mass 14 is restricted from rotational movement within thechamber 12, e.g., by the damper 24 serving as a rotationally rigid linkbetween the stabilizer mass 14 and the chamber 12. However, it may bedesired to provide for rotational movement of the stabilizer mass 14relative to the chamber 12, adjusting the direction and speed of therotation in accordance with any torsional vibration that may occur alongthe drill stem. This may be accomplished by installing a drive motor 34at some convenient location between the structure of the chamber 12 andthe stabilizer mass 14, e.g., affixing the motor 34 to the upper portion26 of the chamber 12 and connecting the motor 34 to the damper 24 toselectively rotate the damper 24. Alternatively, the motor 34 might becombined structurally with the damper 24. In any event, appropriatepower input to the motor 34 rotates or torsionally oscillates thestabilizer mass 14 accordingly to dampen or eliminate torsionalvibrations during drilling operations.

FIG. 2 of the drawings provides a schematic illustration of theinstallation of the attenuator 10 of FIG. 1 within a drill string, ormore specifically within the bottom hole assembly (BHA) of such a drillstring. The BHA is conventionally that portion of the drill string atthe lower end or bottom of the drill string. The drill bit, designatedDB in the drawing, is attached to the lower end of the BHA. The BHAcomprises a hollow length of drill pipe (designated DP in the drawing),and may have additional electronic or other devices installed therein totransmit information to the surface. The hollow drill pipe DP has aninner surface S defining its open core C. The attenuator 10 may beinstalled concentrically within the drill pipe DP. The attenuator 10 hasa smaller span or diameter than that of the open core C, so that theouter surface 36 of the attenuator 10 and the inner surface S of thedrill pipe DP define a toroidal passage P or attenuator clearance volumetherebetween. Thus, the attenuator 10 remains free of contact with theinner surface S of the drill pipe DP. The attenuator 10 is preferablyimmovably affixed within the drill pipe DP, i.e., restricted fromrotation relative to the drill pipe DP.

FIG. 3A of the drawings is a schematic diagram of a drill string anddrive table illustrating various parameters affected by the installationand performance of the attenuator 10, while FIG. 3B is a similar diagramillustrating parameters associated with the attenuator 10. FIG. 4A isanother schematic diagram illustrating the foregoing parameters affecteddynamically by use of the attenuator 10. FIG. 4B shows a diagrammaticrepresentation of a polycrystalline-diamond-compact (PDC) drill bit,illustrating various parameters of concern when the attenuator 10 isused with a PDC bit.

Before proceeding further with a description of the attenuator 10, andparticularly the various graphs and charts provided to illustrate thevarious parameters involved and their effects on the efficiency of theattenuator 10, it is appropriate to provide tables listing the variousparameters, terms and symbols used herein and their definitions. Thevalues shown in the right hand columns are purely exemplary, and may beadjusted as required.

TABLE 1 Parameter Description and Exemplary Values Parameter DefinitionExemplary Value BHA Bottom Hole Assembly N/A DDE Delay DifferentialEquation N/A PDC Polycrystalline-Diamond-Compact bit N/A ROP Rate ofPenetration (of bit into N/A substrate) TOB Torque on Bit duringdrilling N/A operations WOB Weight on Bit during drilling N/A operationsJ Drill Collar Inertia 415 kg · m² m_(d) Drill Collar Mass 87000 kgK_(T) Equivalent Drill Pipe Torsional 600 N · m/rad Stiffness C_(T)Added Drill Collar Torsional 500 N · s²/rad Damping W_(o) Static Load100 kN A Drill Bit Radius 0.15 m l Wearflat Length 5 mm σ Average NormalStress 112 MPa E Intrinsic Specific Energy 160 MPa ξ Inclination ofCutting Force on 0.7 Cutting Face Γ Spatial Orientation of Wearflats 1.2μ₀ Coefficient of Friction 0.06 n Number of Blades 8 m_(f) AttenuatorMass Variable c_(f) Attenuator Damping Variable k_(f) AttenuatorStiffness Variable ω_(d) Rotary Speed of Rotary Table Variable

TABLE 2 Simulation Parameters (Roller-Cone Drill Bits) and ExemplaryValues Parameter Definition Exemplary Value k_(c) Rock FormationStiffness 67 MN/m s_(o) Formation Elevation Amplitude 1 mm B FormationSurface Function Constant 1 c₁ Penetration Constant 1.35e−8 c2Penetration Constant −1.9e−4

FIG. 3A is a schematic diagram of a drill string, illustrating variousparameters relevant to vibration of the drill string. In FIG. 3A, theBHA is represented by a lower cylinder having closely spaced horizontallines thereacross, and the drill pipe and its elasticity are shownschematically by the spring symbol between the top of the BHA and thetop drive of the assembly. The equivalent drill pipe torsional stiffnessand added drill collar torsional damping of the drill pipe are indicatedby the symbols K_(t) and C_(t), respectively, as shown in Table 1further above. The Weight on Bit (WOB) is shown by a vertical arrow atthe bottom of the BHA, and the Torque on Bit (TOB) is shown by anelliptical representation of the circular path of the drill assemblyduring operation. The symbols φ and Ω represent torsional factorsarising during the drilling process, and X_(d) represents the depth ofthe hole from the surface to the bottom of the hole. The top drive isthe prime mover that applies the torque to rotate the drill stem and itsBHA and DB in the hole. The symbol W₀ represents the static load on thesystem, and the symbols ω_(d) represents the rotational drive speed ofthe rotary top drive table. T_(d) and ω_(d)t represent additionalfactors.

Prior to the installation of the attenuator 10 in the BHA of the drillstring, the torsional equation of motion is given as:

J{umlaut over (φ)}+C _(T) {dot over (φ)}+K _(T) φ=T _(d)−TOB  (1)

where

T _(d) =K _(T)·ω_(d) ·t.  (2)

The axial equation of motion is given as:

m _(DC) {umlaut over (x)} _(d) =W ₀−WOB.  (3)

The depth of cut per revolution per blade is given as:

d _(n)(t)=x _(d)(t)−x _(d)(t−t _(n)).  (4)

The total depth of cut is given as:

d=n·d _(n)  (5)

where t_(n) is the instantaneous time delay obtained by solving theequation:

φ(t)−φ(t−t _(n))=2π/n.  (6)

The WOB and the TOB both have cutting and friction components:

WOB=W _(c) +W _(f);  (7)

TOB=T _(c) +T _(f).  (8)

The expressions for the cutting components are given as:

$\begin{matrix}{{W_{c} = {a \cdot \xi \cdot ɛ \cdot d}};} & (9) \\{T_{c} = {\frac{a^{2}}{2} \cdot ɛ \cdot {d.}}} & (10)\end{matrix}$

The friction components are given as:

$\begin{matrix}{{W_{f} = {a \cdot l \cdot \sigma}};} & (11) \\{T_{f} = {\frac{a^{2}}{2} \cdot \gamma \cdot l \cdot \sigma \cdot {\mu.}}} & (12)\end{matrix}$

When the drill bit loses contact with the formation during bit bounce,the depth of cut, d, is negative and the friction components vanish.

By substituting Equations (10) and (12) into equation (1), we have:

$\begin{matrix}{{{J\overset{¨}{\varphi}} + {C_{T}\overset{.}{\varphi}} + {K_{T}\phi} + {\frac{a^{2}}{2} \cdot ɛ \cdot n \cdot {x_{d}(t)}}} = {T_{d} - T_{f} + {\frac{a^{2}}{2} \cdot ɛ \cdot n \cdot {{x_{d}\left( {t - t_{n}} \right)}.}}}} & (13)\end{matrix}$

By substituting Equation (9) and (11) into Equation (3) we have:

$\begin{matrix}{{{{m_{d}{\overset{¨}{x}}_{d}} + {a \cdot \xi \cdot ɛ \cdot n \cdot {x_{d}(t)}}} = {W_{0} - W_{f} + {a \cdot \xi}}}{\cdot ɛ \cdot n \cdot {{x_{d}\left( {t - t_{n}} \right)}.}}} & (14)\end{matrix}$

With reference to equation (14), the coefficient of x_(d) represents thestiffness of the formation that can be expressed as:

k _(c) =a·ξ·ε·n.  (15)

FIG. 3B is a schematic diagram including parameters that describevibration of the drill string after installation of the drill stringaxial vibration attenuator 10. The various symbols provided in FIG. 3Aare also provided in FIG. 3B, and have identical meanings in the twoFigures. However, FIG. 3B also includes a representation of the sprungmass m_(f), shown as stabilizer mass 14 in FIGS. 1 and 2, and the depthbelow the surface x_(f) of the lower end of the sprung mass m_(f) orstabilizer mass 14, along with the attenuator stiffness k_(f) andattenuator damping c_(f). After adding the attenuator 10 of FIGS. 1 and2 to the assembly represented in FIG. 3A, the equation of motion for thesprung mass m_(f) or stabilizer mass 14 is given as:

m _(f) ·{umlaut over (x)} _(f) +c _(f)({dot over (x)} _(f) −{dot over(x)} _(DC))+k _(f)·(x _(f) −x _(d))=0  (16)

Hence the axial equation of motion of the BHA after adding thestabilizer mass is given as:

m _(d) x _(d) +a·ξ·ε·n·x _(d)(t)=F ₀ −W _(f) +a·ξ·ε·n·x _(d)(t−t _(n))+c_(f)({dot over (x)} _(f) −{dot over (x)} _(d))+k _(f)·(x _(f) −x_(d)).  (17)

The modeling can be enhanced into a finite element model (FEM) byconsidering both the torsional and axial flexibility of the BHA and thedrill pipe. If the displacement vector of the element is denoted byW_(i)=[x_(i) φ_(i) x_(i+1)φ_(i+1)]^(T), and if the moduli of elasticityand rigidity of the element are denoted by E_(i) and G_(i),respectively, the element flexibility matrix K_(i) is given as:

$\begin{matrix}{{K_{i} = \begin{bmatrix}{E_{i}A_{i}\text{/}l_{i}} & 0 & {{- E_{i}}A_{i}\text{/}l_{i}} & 0 \\0 & {G_{i}I_{i}\text{/}l_{i}} & 0 & {{- G_{i}}I_{i}\text{/}l_{i}} \\{{- E_{i}}A_{i}\text{/}l_{i}} & 0 & {E_{i}A_{i}\text{/}l_{i}} & 0 \\0 & {{- G_{i}}I_{i}\text{/}{li}} & 0 & {G_{i}I_{i}\text{/}l_{i}}\end{bmatrix}},} & (18)\end{matrix}$

where A_(i) and I_(i) are the element cross-sectional area and momentarea of inertia, respectively.

As an example, if a 200-meter long BHA is discretized into 20 elementsequal in length and if a 1000-meter drill pipe is added to the FEM, 10additional elements (also equal in length) are appended, providing adrill string having 62 degrees of freedom.

FIG. 4A is another schematic drawing of the drill string illustratingadditional parameters. This model includes coupling between thetorsional and axial motions through the form of drill bit-formationinterface, torque and axial force. Lumped inertias are included in themodel for the axial and torsional motions of the BHA-DC and theabsorber. Also included are spring and damping forces between theabsorber and BHA. The absorber design variables are the absorber's massm_(f), damping c_(f), and stiffness k_(f). The BHA mass and inertia arem_(d) and J, respectively. The drill pipe torsional stiffness anddamping are K_(r) and C_(r), respectively. The WOB is shown by an axialvector arrow with the TOB shown as an elliptical representation of therotary torque vector, as in FIG. 3B. The symbols φ and Ω representtorsional factors arising during the drilling process, x_(f) representsthe depth below the surface of the lower end of the sprung mass m_(f),and X_(d) represents the depth of the hole from the surface to thebottom of the hole. The top drive is the prime mover that applies thetorque to rotate the drill stem, the BHA, and the DB in the hole. Thesymbol W, represents the static load on the system. The symbol ω_(d)represents the rotational drive speed of the rotary top drive table.T_(d) and ω_(d)t represent additional factors, as indicated in FIG. 3B.

FIG. 4B is a diagrammatic representation of apolycrystalline-diamond-compact (PDC) drill bit, illustrating variousparameters of concern when the drill string axial vibration attenuator10 is used therewith. The numbers 1 and 2 represent two blades orelements of the drill bit. l_(n) represents the wearflat lengthmultiplied by the number of blades, while dn(l) represents the cuttingdepth multiplied by the number of blades. he variables x_(d)(l) andx_(d)(l−l_(n)) represent the vertical depth of the bit and the depth ofthickness of the blades. The bit rotates about the vertical axis z. Thearcuate span of each blade is represented by the term 2π/n. Thevariables Ω, φ(l−l_(n)), and φ(l) represent various rotational factorsinvolved in the bit operation.

Many drilling operations utilize a roller cone drill bit that hasdifferent dynamics than its PDC counterpart, but nevertheless thecoupling between the axial and torsional modes still exists. The WOB inthe case of a roller cone drill bit is given by:

$\begin{matrix}{{WOB} = \left\{ {\begin{matrix}{k_{c}\left( {x_{d} - s} \right)} & {{{if}\mspace{14mu} x_{d}} \geq s} \\0 & {{{if}\mspace{14mu} x_{d}} < s}\end{matrix}.} \right.} & (19)\end{matrix}$

The formation surface elevation, s, is given as:

s=s ₀·sin bφ.  (20)

The torque-on-bit (TOB) is given as:

$\begin{matrix}{{{TOB} = {{WOB} \cdot \left( {{\mu (\varphi)} + \sqrt{\frac{d}{a}}} \right) \cdot a}},} & (21)\end{matrix}$

where a Stribeck friction model is expressed as:

$\begin{matrix}{\mu = {\mu_{0} \cdot {\left( {{\tanh \; \varphi} + \frac{2 \cdot \varphi}{1 + \varphi^{2}} + {0.01\varphi}} \right).}}} & (22)\end{matrix}$

In this case, the depth of cut, d, is given as:

$\begin{matrix}{d = {\frac{2 \cdot \pi \cdot {ROP}}{\omega_{d}}.}} & (23)\end{matrix}$

The rate of penetration (ROP) is defined as:

ROP=c ₁ ·F ₀·√{square root over (ω_(d))}+c ₂.  (24)

The description and exemplary values of the parameters for the PDC androller cone drill bit cases are listed in tables 1 and 2, respectively,further above.

Equations 13 and 14 (or 16 and 17, after adding the attenuator)represent a system of delay differential equations (DDEs) that can beexpressed in a state-space model with a time delay τ either constant orvariable as:

{dot over (x)}(t)=A ₀ ·x(t)+A ₁·(t−τ)+B·u(t).  (25)

The characteristic equation is a quasi-polynomial in the form of:

|s·I−A ₀ −A ₁ ·e ^(−τ·s)|  (26)

The presence of the term e^(−τ·s) leads to a theoretical infinite numberof complex solutions for a continuous system. Here, the ChebyshevSpectral Method is presented as a numerical method to solve DDEs of adiscrete system that will have a finite number of roots, since a closedform solution is virtually impossible to obtain. In the present system,the quasi-polynomial is solved using the approach developed by Breda etal. to discretize the solution operator. The solution operator is theoperator transforming an initial condition φ onto the solution segmentat a later timepoint specified by a parameter h, in the following sense.

We define the solution operator of the DDE in equation (25) to be theoperator transforming an initial condition φ to the solution segment attimepoint h. This operator is denoted by T(h):C([−τ, 0],

^(n))→C([−τ, 0],

^(n)). The solution operator applied to φ, i.e., (T(h)φ)(θ)=: ψ(θ), isthe solution segment of (2.1) with initial condition φ=φ=at timepoint h.More precisely, ψ(θ):=(T (h)φ)(θ)=x(h+θ), θ∈└−τ, 0┘, where x(t) is thesolution of Equation (25) with initial condition φ=φ.

Every DDE can be rewritten as a partial differential equation (PDE) byintroducing an additional memory-dimension. If the original DDE isrepresented as:

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{x}(t)} = {{A_{0} \cdot {x(t)}} + {A_{1} \cdot {x\left( {t - \tau} \right)}}}} & {t \geq 0} \\{{x(t)} = {\phi (t)}} & {t\; {\varepsilon\left( {{- \tau},0} \right\rbrack}^{\prime}}\end{matrix} \right. & (27)\end{matrix}$

then the equivalent PDE can be written as a boundary value problem as:

$\begin{matrix}\left\{ \begin{matrix}{\frac{\partial u}{\partial\theta} = \frac{\partial u}{\partial t}} & {{t \geq 0},{{\theta\varepsilon}\left\lbrack {{- \tau},0} \right\rbrack}} \\{{u_{\theta}^{\prime}\left( {t,0} \right)} = {{A_{1} \cdot {u\left( {t,{- \tau}} \right)}} + {A_{0} \cdot {u\left( {t,0} \right)}}}} & {t \geq 0} \\{{u\left( {0,\theta} \right)} = {\phi (\theta)}} & {{\theta\varepsilon}\left\lbrack {{- \tau},0} \right\rbrack}\end{matrix} \right. & (28)\end{matrix}$

for the unknown u∈C(┌0, ∞┐×[−τ, 0],

²). Let φ∈C([−τ, 0],

^(n)) be given. Then if x(t) is the solution to equation (26), and ifu(t, θ) is a solution to equation (28), then:

u(t,θ)=x(t+θ),θ∈[−τ,0],t≧0.  (29)

Let A correspond to the differentiation operator in θ-direction with thedomain of functions fulfilling the boundary conditions in equation (28),that is:

$\begin{matrix}\begin{matrix}{{{\left( {A\; \phi} \right)(\theta)}:={\frac{d\; \phi}{d\; \theta}(\theta)}},} & {{\phi^{\prime}(0)} = {{A_{1}{\varphi \left( {- \tau} \right)}} + {A_{0}{{\phi (0)}.}}}}\end{matrix} & (30)\end{matrix}$

Hence the problem is reduced to an abstract Cauchy-problem in the form:

$\begin{matrix}{{\frac{d}{dt}x_{1}} = {{Ax}_{1}.}} & (31)\end{matrix}$

The differentiation operator, A, of the abstract Cauchy-problem isexpressed in terms of the solution operator of the DDE, T, as:

$\begin{matrix}{{A\; \phi}:={\lim\limits_{t\rightarrow 0^{+}}{\frac{1}{t}{\left( {{{T(t)}\phi} - \phi} \right).}}}} & (32)\end{matrix}$

The eigenvalues of the operator A are the eigenvalues of the DDE. It isnow required to discretize A and compute the eigenvalues of thecorresponding finite-dimensional linear operator A_(n) that representsthe eigenvalues of the system with the time-delay incorporated.

For a given natural number, N, the Chebyshev nodes over the interval[−τ, 0] are defined as:

$\begin{matrix}\begin{matrix}{x_{N,i} = {\frac{\tau}{2} \cdot \left( {{\cos \left( {i \cdot \frac{\pi}{N}} \right)} - 1} \right)}} & {{i = 0},1,2,\cdots \mspace{11mu},{N.}}\end{matrix} & (33)\end{matrix}$

The Chebyshev differentiation matrix, D_(N), is obtained utilizing theChebyshev nodes as:

D N = - 2 τ  ( - 1 1 ⋱ ⋱ - 1 1 ) ∈ N × ( N + 1 ) . ( 34 )

Thus, A_(N) can now be evaluated as:

A N = [ D N ⊗ I n A 1  0   ⋯   0  A 0 ] ∈ mN × mN , ( 35 )

where m is the number of degrees of freedom of the original state spaceof the continuous DDE.

At a certain operating point there exists a minimum stability speed ofthe top drive of the rotary table, below which the system becomesunstable due to increased time delay due to the cutting action of theblades of the PDC drill bit. It is well known that time delay resemblesnegative damping that destabilizes a system. Hence, the BHA will be moresusceptible to bit bounce when the speed of the top drive is lowered. Indrilling operations, there are several reasons as to why the top drivespeed is decreased. Logging-while-drilling (LWD) ormeasurement-while-drilling (MWD) operations usually require lowerpenetration rates in order to obtain accurate data of the well. Anotherreason is the power limitation on the motor driving the rotary table,since the torque load on the driving motor increases as the friction atthe drill bit increases and puts a limit on the maximum rotary speed.Even during normal operation, with increasing torque on bit (TOB), therotary speed of the drill bit would be oscillating around the mean valueof the top drive speed, and when its speed falls significantly, the timedelay between the cutting actions of the individual blades willincrease. Moreover, the addition of new sections of drill pipe requiresthat the drilling action be temporarily paused, bringing the rotarytable to a complete stop and then restarting and passing through low RPMranges that are susceptible to bit bounce.

One of the methods for stabilizing the system is by adding additionalmass to the BHA in order to suppress bit bounce. This can be done byeither increasing the BHA length, which is referred to as the “unsprungmass” case, or by attaching attenuator units comprising a mass sprung onone or more springs and dampers, generally as shown schematically inFIGS. 1 and 2. FIG. 5A is a graph showing the minimum mass ratiorequired to suppress bit bounce over a top drive speed ranging from 65to 130 RPM at a rock stress of 160 MPa and formation stiffness of 134MPa. At speeds of 130 RPM or higher the time delay is small enough andthe system is stable, so no added mass is needed. It can be clearly seenthat the sprung added mass needed is much smaller than its unsprungcounterpart. A rigorous and controlled search using computerizednumerical integration was carried out to arrive at the respectivestiffness and damping values of the spring and damper that yielded theminimum and optimum sprung mass required to suppress bit bounce. FIG. 5Bis a three dimensional plot illustrating the corresponding sprung massvalues. The optimum mass values are shown in bold face. The optimum masswas chosen to be the minimum value at each rotational speed. It is to benoted that the entire search elapsed over twenty hours of computationaltime, resulting in 120,000 possible search points.

Consider a case where the top drive speed decreases to 105 RPM. Bitbounce vibrations will be seen, as shown clearly in the graph of FIG.6A, and there also will be large fluctuations of the BHA rotaryvelocity, as shown in the corresponding graph of FIG. 6B. In these twographs the intrinsic specific energy ε is 160 MPa, and the formationstiffness is 134 MPa. In this case, bit bounce suppression requires thatthe BHA mass be increased by 49 tons if the mass is fixed to the BHA,i.e., as unsprung mass. However, the added mass need only be increasedby 17 tons if the additional mass is attached to the BHA by a spring 18having appropriately selected stiffness, e.g., 42.4 MN/m, and a damper24, e.g., 0.72 MN.s/m, to provide the sprung mass. The same result isachieved when the sprung mass is about three times less than itsunsprung counterpart. FIGS. 7A and 7B are graphs respectivelyillustrating the axial and torsional velocities of the BHA after addingthe attenuator, with ε=160 MPa and formation stiffness=134 MPa, as inother examples above.

Since numerical integration of the dynamic equations of motion is verytime-consuming, and thus a relatively small number of possible solutionscan be scanned through trial and error, the Chebyshev-baseddiscretization spectral method was implemented in order to find valuesof the attenuator mass, spring stiffness, and damping that achieveoverall system stability. By applying equation (35) above one can obtaina characteristic matrix A_(N), in which the eigenvalues determine thestability. The system is stable if, and only if, all eigenvalues havenegative real values. Hence, the computational time is reduced, andthousands of possible solutions can be scanned in just a matter ofminutes. FIG. 8 is a three-dimensional plot of solutions obtained usingthe Chebyshev-based discretization spectral method. The number of searchpoints was approximately ten million, yet the elapsed search time wasless than four hours. If this search were to be conducted manually usingnumerical integration, the search time would be 58 days. If the drillstring model were finite element-based, then at least one thousand dayswould be required to conduct this search.

The validity of the Chebyshev method is verified by numericalintegration of the equations of motion under the same conditions.Consider a case when the equivalent soil stiffness is 67 MPa and the topdrive speed is 80 RPM. Without the attenuator 10, the system isunstable, as can be seen in FIG. 9, where there exist oscillations ofaxial velocity. The Chebyshev method predicts this instability throughthe evaluation of eigenvalues, and it can be seen in FIG. 10A that thereare two conjugate and complex poles on the imaginary axis. Note that inFIG. 10A the rightmost conjugate poles have a frequency of 31 rad/s. Theperiodic time from FIG. 9 is approximately 0.2 seconds, whichcorresponds to an oscillation frequency of 31 rad/s, which is consistentwith the frequency directly evaluated from the eigenvalue plot. Uponinstalling an attenuator that is 15% in mass of the original BHA, andwhich has a natural frequency and damping ratio of 30 rad/s and 0.3,respectively, the system becomes stable. This is shown by the graph ofFIG. 10B, showing the eigenvalue plot obtained by the Chebyshev methodand indicating that the previously unstable poles have migrated into theleft-hand plane, further validating the spectral method implementedhere.

Another advantage of the Chebyshev method is that it can be implementedto obtain or predict an operating point stability chart. FIG. 11 is sucha chart, showing a coarse mesh of stable operating points that indicatestable top drive spin speeds over a range of values of rock formationstiffness. The larger open ovals shown on the chart of FIG. 11 are onthe stability boundary line, in which a small decrease in top drive spinspeed or increase in formation stiffness value will result in bit bouncein the system. This is verified by numerical integration of theequations of motion by perturbing about the three different boundaryline stable points shown by the larger open ovals. FIG. 12A shows theaxial velocity of the drill bit at the first point of investigation,where the top drive spin speed is 78 RPM and the formation stiffness is53 MPa. In this case, the axial velocity reaches a steady-state valuewith the system being stable, as the FIG. 12A graph indicates. Either areduction of top drive rotational speed or an increase in formationstiffness would be destabilizing, as noted further above. In the graphof FIG. 12B the rotational velocity of the top drive, i.e., 78 RPM, isthe same as that of the graph of FIG. 12A. However, the rock formationstiffness in FIG. 12B is 55 PMa, slightly higher than the 53 MPa of FIG.12A. This is sufficient to cause some destabilization, as indicated inFIG. 12B.

FIGS. 13A and 13B are additional graphs showing the destabilizingeffects of increases in rock formation stiffness. In FIG. 13A, the drillbit spin velocity has been increased to 105 RPM, but this isinsufficient to offset the destabilizing influence of the increase inrock formation stiffness to 100 MPa, as opposed to the 53 to 55 MPaformation stiffnesses of the graphs of FIGS. 12A and 12B. The graph ofFIG. 13B shows the results of operation at a rotational velocity of 135RPM, normally a stable speed in softer formations, but operating in arock formation having a stiffness of 154 MPa. This combination resultsin axial instability or bit bounce, as indicated by the FIG. 13B graph.Similarly, the graph of FIG. 13C shows the resulting instability at atop drive rotational velocity of 140 RPM and a rock formation stiffnessof 158 MPa.

FIG. 14 illustrates a very fine mesh stability chart showing top drivespeed v. rock formation stiffness for different attenuator mass values.The chart was divided into 1.2 million search points, and the elapsedtime to evaluate this chart was only forty-five minutes. In the chart ofFIG. 14, the attenuator mass values (indicated as PPMD, or Passive ProofMass Damper, in the legend of FIG. 14) increase from zero in thelowermost band up to an increase of 40% in the larger shaded area of theupper portion of the chart. The shading of the various bands indicatingdifferent percentages of mass increase correspond to the shadings andcorresponding mass increases shown in the small circular areas of thelegend.

FIGS. 15A through 15D are graphs showing the system performance beforeadding the attenuator 10, where the system stiffness is 100 MPa. Thesegraphs are based upon a lumped model and finite element modeling (FEM).FIG. 15A is a graph of the axial velocity near the drill bit, FIG. 15Bis a graph of the torsional velocity near the drill bit, FIG. 15C is agraph of the drill bit axial reaction force (WOB), and FIG. 15D is agraph of the drill bit torsional reaction force (TOB). In each of thesedrawings, the lumped model is shown by the heavier continuous line, theFEM for the drill string is shown by the lighter continuous line, andthe FEM for the BHA is shown by the broken line. The continuousinstability that continues completely across each of these graphs isapparent.

In contrast, FIGS. 16A through 16D are graphs showing the systemperformance after adding the attenuator 10, where the system stiffnessis identical to that of FIGS. 15A through 15D at 100 MPa. The mass ofthe added attenuator is 15% of the original mass of the BHA in theseexamples, and has a natural frequency and damping ratio of 30 rad/s and0.3, respectively. The graphs of FIGS. 16A through 16D are also basedupon a lumped model and finite element modeling (FEM). FIG. 16A is agraph of the axial velocity near the drill bit, FIG. 16B is a graph ofthe torsional velocity near the drill bit, FIG. 16C is a graph of thedrill bit axial reaction force (WOB), and FIG. 16D is a graph of thedrill bit torsional reaction force (TOB). In each of these Figs., thelumped model is shown by the heavier continuous line, the FEM for thedrill string is shown by the lighter continuous line, and the FEM forthe BHA is shown by the broken line. While there is not a significantdifference between the FEM of the complete drill string and the FEM ofonly the BHA in any of FIGS. 15A through 16D, it will be seen that theaddition of the attenuator 10 rapidly attenuates the vibrationalfrequencies as time progresses, as shown by the broken linesrepresenting the FEM BHA in each of FIGS. 16A through 16D. The onsetinstability speed of approximately 95 RPM is nearly the same in allthree cases, i.e., lumped model, FEM drill string, and FEM BHA, whichindicates that the lowest modes of vibration are excited as describedfurther above.

In order to demonstrate robustness, consider a universal attenuator witha natural frequency of 30 rad/s and a damping ratio of 0.3 to be fittedupon any BHA that utilizes either a PDC or roller cone drill bit. FIG.17A is a graph illustrating the decrease in axial vibration amplitudeswith increasing attenuator mass for different operating top drive spinvelocities when a PDC drill bit is used. FIG. 17B is similar, but showsthe results when a roller cone drill bit is used. FIGS. 18A and 18B areadditional graphs illustrating a comparison between vibration amplitudevalues for added sprung and unsprung masses at a top drive spin speed of80 RPM, with FIG. 18A being for a PDC bit and FIG. 18B being for aroller cone bit.

FIGS. 18C and 18D are graphs illustrating vibration amplitude values v.rock formation stiffness when an attenuator having 15% total sprung andunsprung mass is added to the original exemplary 87-ton BHA and the topdrive spinning at 80 RPM. FIG. 18C is for a PDC drill bit, while FIG.18D is for a roller cone bit.

To demonstrate the effectiveness and advantages of the attenuator overstandard or conventional shock absorbers, consider a case where the soilformation is not sufficiently hard to induce bit bounce vibrations at agiven top drive spin speed, and hence no axial vibration suppressiondevice is needed. Equation 36 below represents a simplified model of astandard shock absorber with stiffness k_(abs) and damping c_(abs). Foursimulations are conducted, including (i), original BHA, (ii) tunedabsorber, (iii) mistuned absorber, and (iv) with the attenuator 10. Theequivalent rock formation stiffness is 150 MN/m, and the top drive speedis 150 RPM, at which there is no bit bounce at steady-state speed.Adding an attenuator of mass ratio 0.15, natural frequency of 30 rad/s,and damping ratio of 0.3, or a tuned shock absorber with stiffness valueof 2e4 N/m and damping value of 30e3 N.s/m, does not affect thestability of the system, as shown in FIGS. 19A and 19B. However, if thestiffness of the shock absorber is increased to 2e5 N/m, the systembecomes unstable, and in addition to bit bounce, stick-slip occurs,since there is a large fluctuation of both torsional and axialvelocities. Thus, this demonstration confirms that adding shockabsorbers may exacerbate the vibration, rather than mitigating suchvibration.

$\begin{matrix}{F_{abs} = \left\{ \begin{matrix}0 & {{\overset{.}{x}}_{DC} \leq 0} \\{{{- k_{abs}} \cdot x_{DC}} - {c_{abs} \cdot {\overset{.}{x}}_{DC}}} & {{\overset{.}{x}}_{DC} > 0}\end{matrix} \right.} & (36)\end{matrix}$

Now consider the case when drilling into a harder formation ofequivalent rock formation stiffness of 170 MN/m with the same top drivespin speed of 150 RPM. In this case, severe bit bounce and torsionalvibrations occur that nearly force the system into stick-slip. As shownin FIGS. 20A and 20B, adding the attenuator (passive proof mass damper,or PPMD) suppresses the bit bounce and torsional vibrations, and usingthe conventional “tuned” shock absorber worsens the vibration, since thesystem experiences stick-slip as the drill bit velocity slows to zeroRPM. Thus, the robustness of using the attenuator 10 is furthervalidated by comparison with a conventional shock absorber operatingunder identical conditions.

FIGS. 21 through 24 are graphs illustrating further variations of theattenuator, drill string, and BHA assembly. Another approach to theattenuator configuration is to increase the attenuator mass whilereducing the BHA and/or DC mass in order to maintain a constant totalmass of the assembly. The graph of FIG. 21 illustrates the amount ofattenuator mass (as a percentage of the exemplary 87-ton total mass)required to suppress bit bounce vs. top drive spin speed. This FIG. 21graph corresponds to the use of a bladed PDC bit, but the attenuator isalso effective in suppressing axial BHA vibrations when a roller conebit is used, as noted further above.

FIG. 22 is a graph illustrating the undamped axial vibrations that occurin a formation having a rock formation stiffness of 67 MN/m and anundamped 87-ton BHA, i.e., the same total weight as the attenuator andBHA combination used to form the graph of FIG. 21, and a top driverotational speed of 76 RPM. As can be seen in FIG. 22, the axialvibrations remain substantially constant over a given period of timewith no damping whatsoever. By attaching an attenuator weighing 20% ofthe original weight of the BHA and having a natural frequency of 30rad/s and damping ratio of 0.3, the vibrations are mitigated over aperiod of time, as indicated in the graph of FIG. 23.

Finally, consider an absorber with a natural frequency of 30 rad/s and adamping ratio of 0.3 that may be fitted to any BHA using either a PDC orroller cone drill bit. FIG. 24 shows the decrease of axial vibrationamplitudes with increasing absorber mass for different operating topdrive spin velocities when a PDC drill bit is used. This is similar tothe results when using a roller cone drill bit, as shown in the graph ofFIG. 17B further above.

In conclusion, a passive proof mass damper or attenuator serves tomitigate or fully suppress harmful axial vibrations in a drill stringBHA. Simulation studies indicate that the sprung mass of the attenuatoreffectively mitigates (and can in some cases completely suppress) bitbounce, and that the value of the sprung mass is much less than thecorresponding unsprung mass required to yield the same effect. Thispermits the driller to operate the top drive with minimal spin speed,thus conserving power and operational costs. Moreover, the spring anddamper of the attenuator can be fine-tuned in order to minimize thesprung mass. The robustness of the attenuator is illustrated herein fora wide range of operating conditions for both PDC and roller cone drillbit models, as well as in comparison to a standard shock absorberconventionally used for such purposes. One, two, or more suchattenuators may be installed in the drill string, particularly in theBHA of the drill string. Furthermore, the Chebyshev-based discretizationspectral method has been applied in order to predict system stability byestimating eigenvalues and to estimate stabilizing attenuator parametersin a very time efficient manner, instead of the time-consuming trial anderror process performed using numerical integration. Verification of theChebyshev method DDE solver has been confirmed by comparison with thenumerical integration of the equations of motion utilizing both a lumpedmodel and FEM.

It is to be understood that the present invention is not limited to theembodiments described above, but encompasses any and all embodimentswithin the scope of the following claims.

We claim:
 1. A drill string axial vibration attenuator, comprising: anelongate sealed chamber adapted for being immovably affixed within apipe of a drill string; and an elongate stabilizer mass resilientlysupported within the sealed chamber.
 2. The drill string axial vibrationattenuator according to claim 1, wherein: the chamber is adapted forbeing disposed concentrically within one of the pipes of the drillstring, the chamber having an outer surface and an inner surface, thechamber outer surface and the inner surface of the pipe defining atoroidal passage therebetween; and the stabilizer mass is disposedconcentrically within the sealed chamber, the stabilizer mass having anouter surface, the stabilizer mass outer surface and the chamber innersurface defining a toroidal stabilizer mass clearance volumetherebetween, the stabilizer mass being free of contact with the innersurface of the chamber.
 3. The drill string axial vibration attenuatoraccording to claim 1, wherein the chamber has a lower end and an upperend opposite the lower end and the stabilizer mass has a lower end andan upper end opposite the lower end, the attenuator further comprising:a compression spring disposed between the lower end of the chamber andthe lower end of the stabilizer mass; and a damper disposed between theupper end of the chamber and the upper end of the stabilizer mass. 4.The drill string axial vibration attenuator according to claim 3,wherein the compression spring is selected from the group consisting ofsprings having linear spring rates and springs having nonlinear springrates.
 5. The drill string axial vibration attenuator according to claim3, wherein the damper is selected from the group consisting of dampershaving linear damper rates and dampers having nonlinear damper rates. 6.The drill string axial vibration attenuator according to claim 1,wherein the stabilizer mass is rotationally stationary relative to thechamber.
 7. The drill string axial vibration attenuator according toclaim 1, further comprising a motor communicating with the stabilizermass, the motor selectively rotating the stabilizer mass.
 8. A drillstring axial vibration attenuator, comprising: an elongate sealedchamber adapted for being disposed concentrically within a pipe of adrill string, the chamber having an outer surface and an inner surface,the chamber outer surface and the inner surface of the pipe defining atoroidal passage therebetween, the chamber having an upper end and alower end; a first resilient member extending downward within thechamber from the upper end; a second resilient member extending upwardwithin the chamber from the lower end; and an elongate stabilizer massdisposed concentrically within the sealed chamber, the stabilizer massbeing attached to and suspended between the first and second resilientmembers, the stabilizer mass having an outer surface, the stabilizermass outer surface and the chamber inner surface defining a toroidalstabilizer clearance volume therebetween, the stabilizer being free ofcontact with the inner surface of the chamber.
 9. The drill string axialvibration attenuator according to claim 8, wherein: the chamber isimmovably affixed within one of the pipes of the drill string; and thestabilizer mass is resiliently supported within the sealed chamber. 10.The drill string axial vibration attenuator according to claim 8,wherein the stabilizer mass has a lower end and an upper end oppositethe lower end, the attenuator further comprising; a compression springdisposed between the lower end of the chamber and the lower end of thestabilizer mass; and a damper disposed between the upper end of thechamber and the upper end of the stabilizer mass.
 11. The drill stringaxial vibration attenuator according to claim 10, wherein thecompression spring is selected from the group consisting of springshaving linear spring rates and springs having nonlinear spring rates.12. The drill string axial vibration attenuator according to claim 10,wherein the damper is selected from the group consisting of dampershaving linear damper rates and dampers having nonlinear damper rates.13. The drill string axial vibration attenuator according to claim 8,wherein the stabilizer mass is rotationally stationary relative to thechamber.
 14. The drill string axial vibration attenuator according toclaim 8, further comprising a motor communicating with the stabilizermass, the motor selectively rotating the stabilizer mass.
 15. A drillstring with axial vibration attenuation, comprising: a plurality ofelongate pipes connected end-to-end to define a drill string, the drillstring having an upper end and a lower end; a drill bit mounted on thelower end of the drill string; a bottom hole assembly located in thelower end of the drill string; and at least one attenuator mounted inthe bottom hole assembly, the attenuator having: an elongate sealedchamber mounted coaxially within the bottom hole assembly, the chamberbeing annularly spaced from the bottom hole assembly, the chamber havingan upper end and a lower end; a vibration damper mounted at the upperend of the chamber; a spring mounted at the lower end of the chamber;and an elongate mass having an upper end attached to the damper and alower end attached to the spring, the elongate mass being annularlyspaced from the chamber and resiliently move upward and downward tocounteract axial forces occurring during drilling operations, therebypreventing and reducing axial vibration of the drill string.
 16. Thedrill string with axial vibration attenuation according to claim 15,wherein the chamber is immovably affixed within one of the pipes of thedrill string.
 17. The drill string with axial vibration attenuationaccording to claim 15, wherein: the chamber has an outer surface and aninner surface, the chamber outer surface and the inner surface of thepipe defining a toroidal passage therebetween; and the elongate mass isdisposed concentrically within the chamber, the elongate mass having anouter surface, the elongate mass outer surface and the chamber innersurface defining a toroidal stabilizer clearance volume therebetween,the elongate mass being free of contact with the inner surface of thechamber.
 18. The drill string with axial vibration attenuation accordingto claim 15, wherein: the spring is selected from the group consistingof springs having linear spring rates and spring having nonlinear springrates; and the vibration damper is selected from the group consisting ofvibration dampers having linear damper rates and vibration dampershaving nonlinear damper rates.
 19. The drill string with axial vibrationattenuation according to claim 15, wherein the elongate mass isrotationally stationary relative to the chamber.
 20. The drill stringwith axial vibration attenuation according to claim 15, furthercomprising a motor communicating with the elongate mass, the motorselectively rotating the elongate mass.